On the numerical evaluation of fractional Sobolev norms

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September 2007, 6(3): 587-605. doi: 10.3934/cpaa.2007.6.587

On the numerical evaluation of fractional Sobolev norms

1.	Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station, C0200, Austin, TX 78712, United States

Received February 2006 Revised February 2007 Published June 2007

Abstract
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In several important and active fields of modern applied mathematics, such as the numerical solution of PDE-constrained control problems or various applications in image processing and data fitting, the evaluation of (integer and real) Sobolev norms constitutes a crucial ingredient. Different approaches exist for varying ranges of smoothness indices and with varying properties concerning exactness, equivalence and the computing time for the numerical evaluation. These can usually be expressed in terms of discrete Riesz operators.
We propose a collection of criteria which allow to compare different constructions. Then we develop a unified approach which is valid for non-negative real smoothness indices for standard finite elements, and for positive and negative real smoothness for biorthogonal wavelet bases. This construction delivers a wider range of exactness than the currently known constructions and is computable in linear time.

Keywords: control problem, finite elements, biorthogonal wavelets., Fractional Sobolev norm, elliptic PDE.

Mathematics Subject Classification: Primary: 65K10, 65J10; Secondary: 65T6.

Citation: Carsten Burstedde. On the numerical evaluation of fractional Sobolev norms. Communications on Pure & Applied Analysis, 2007, 6 (3) : 587-605. doi: 10.3934/cpaa.2007.6.587

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